WASHINGTON — Senator Dianne Feinstein on Wednesday pushed back on calls for her resignation but asked to step away from the Judiciary Committee indefinitely while recovering from shingles, responding to mounting pressure from Democrats who have publicly vented concerns that she is unable to perform her job.

Ms. Feinstein, an 89-year-old California Democrat, has been away from the Senate since February, when she was diagnosed with the infection. Her absence has become a problem for Senate Democrats, limiting their ability to move forward with judicial nominations. In recent days, as it became clear she was not planning to return after a two-week recess, pressure began to increase for Ms. Feinstein to resign.

On Wednesday night, she said she would not do so, but offered a stopgap solution, saying she would request a temporary replacement on the panel.

“I understand that my absence could delay the important work of the Judiciary Committee,” Ms. Feinstein said in a statement on Wednesday night, after two House Democrats publicly called on her to leave the Senate. “So I’ve asked Leader Schumer to ask the Senate to allow another Democratic senator to temporarily serve until I’m able to resume my committee work.”

In a statement, a spokesman for Senator Chuck Schumer, Democrat of New York and the majority leader, said that Mr. Schumer would make that request of the Senate next week.

Replacing Ms. Feinstein on the committee would require Democrats to pass a resolution, which would need some degree of bipartisan support — either the unanimous consent of the Senate or 60 votes. It is not clear whether Republicans, who want to hold up President Biden’s judicial nominations, would support such a measure.

Ms. Feinstein has missed 58 Senate votes since February, and Democrats did not want to head into the spring and summer without the ability to move ahead on judicial nominations. Under the Senate’s current rules, a tie vote on a nomination in the committee means it fails and cannot be brought to the floor.

“I’m anxious, because I can’t really have a markup of new judge nominees until she’s there,” Senator Richard J. Durbin, Democrat of Illinois and the chairman of the Judiciary Committee, told Politico last month.

Ms. Feinstein, whose growing memory and cognitive issues have prompted mounting concerns among her colleagues, announced earlier this year that she would not seek re-election in 2024, but said she intended to finish out her term.

But her most recent health setback has pushed at least one Democrat from her home state to demand that she immediately step down.

“We need to put the country ahead of personal loyalty,” Representative Ro Khanna, Democrat of California, tweeted on Wednesday, calling for her resignation. “While she has had a lifetime of public service, it is obvious she can no longer fulfill her duties. Not speaking out undermines our credibility as elected representatives of the people.”

A second House Democrat, Representative Dean Phillips of Minnesota, echoed the sentiment in a posting on Twitter, saying Ms. Feinstein remaining in the Senate was a “dereliction of duty.”

The Judiciary Committee has planned a packed spring schedule. Mr. Durbin has announced a full committee hearing to examine the fallout of the Supreme Court’s decision to overturn Roe v. Wade. He has also promised action in response to a ProPublica report that Justice Clarence Thomas has for years accepted luxury travel on yachts and private jets from a conservative donor that he failed to disclose. Much of that work could be complicated by Ms. Feinstein’s indefinite absence if Republicans mount challenges and Democrats need votes to hold them off.

Ms. Feinstein on Wednesday acknowledged that her recovery was taking longer than she had anticipated.

“When I was first diagnosed with shingles, I expected to return by the end of the March work period,” she said. “Unfortunately, my return to Washington has been delayed due to continued complications related to my diagnosis.”

She said she still planned to return to the Senate “as soon as possible,” once her doctors cleared her for travel. And she made it clear that she did not intend to resign.

“I remain committed to the job and will continue to work from home in San Francisco,” she said.

Her desire to hang on to her post, despite her health and memory issues, has irked many of her colleagues on the left. But Ms. Feinstein has previously agreed to sideline herself amid concerns from her party.

In 2020, she agreed to relinquish the top Democratic spot on the Judiciary Committee amid mounting pressure from progressives who said she was not up to the task of leading a crucial panel at the forefront of the partisan war over the courts in a new Biden administration.

Carl Hulse contributed reporting.